生醫應用上之蠕動式微幫浦性能探討

國立交通大學

機械工程學系

碩士論文

生醫應用上之蠕動式微幫浦性能探討

Analysis of the Performance on Peristaltic

Micropumps for Biomedical Applications

研究生 ：關 恕

指導教授 ：陳俊勳 教授

生醫應用上之蠕動式微幫浦性能探討

Analysis of the Performance on Peristaltic Micropumps for Biomedical Applications 研究生 ：

關 恕

Student：Kuan Shu 指導教授：陳俊勳 Advisor：Chiun-Hsun Chen 國 立 交 通 大 學 機 械 工 程 學 系 碩 士 論 文 A Thesis

Submitted to Department of Mechanical Engineering College of Engineering

National Chiao Tung University In Partial Fulfiillment of the Requirements

For the Degree of Master of Science In Mechanical Engineering

June 2006

Hsinchu, Taiwan, Republic of China

)

Analvsis of the Performance on Peristaltic Micropumps for

Biomedical Application

生醫應用上之蠕動式微幫浦性能探討 學生：關 恕 指導教授：陳俊勳 國立交通大學機械工程學系碩士班 摘要 本論文描述利用微機電系統技術的壓電無閥門蠕動式微型幫浦的 設計、製程與應用。微幫浦驅動原理在於 PZT 受到驅動訊號使得致動薄 膜做往覆週期的運動。如此的裝置就如同活塞提供動力驅動微升等級的 流體，如此可利用到許多生醫應用上的實驗室晶片。因此，本論文便是 針對微幫浦的性能分析而探討，主要可分成兩部分。 首先，我們關注的是系統與週遭環境對輸出共振頻率的影響。為提高 蠕動式微型幫浦的輸出能力，從理論上分析了負載流體對致動薄膜動態 特性的影響規律。除了流體的密度以及黏滯性會對附加質量及附加阻尼 造成影響外，流室深度以及驅動相位亦會影響致動薄膜動態特性，造成 頻率位移現象。其次，流體的傳輸能力與電路的設計是具有緊密的連 結。因此藉由適當電路設計的改良能有效提升傳輸流量 1.9~2.8 倍。

Analysis of the Performance on Peristaltic Micropumps for Biomedical Applications

Student：Kuan Shu Advisor：Chiun-Hsun Chen

Department of Mechanical Engineering National Chiao Tung University

Abstract

A valveless peristaltic micropump based on piezoelectric actuation was designed and fabricated. The pump diaphragms are excited by applying pulse signal voltages to three lead zirconate titanate (PZT) disks on glass diaphragm. Such diaphragm structure acts as a “piston” to provide power for the handling of microliter-scaled fluid volumes desired in many lab-on-a-chip chemical and biomedical applications. A high-performance micropump was investigated for this purpose. The paper contains mainly two parts as follows.

First, we are concerned with the interactions between a system and its environment for the influence on output resonance frequency. Analytical results for frequency shift was presented to show that not only does the added mass and added damping depend on both the fluid density and viscosity, the chamber height and actuated signal are as well. Second, the driving circuit can be closely linked with the fluid transport related to the pump performance. Therefore, the improvement design for driving circuit can enhance pump flow rate performance up to 1.9~2.8 times.

Contents

Abstract (in Chinese) ... ………...3

Abstract ………...4

List of figure ...7

List of table ...11

Acknowledgement (in Chinese) ...12

Chapter 1...13

Inefinitesimal Machinery: An Intrigued Word...13

1.1 Introduction………..13

1.2 Literature review………...16

Chapter 2 ……….26

The Damping Effects on the Micro-diaphragm...….26

2.1 Piezoelectric materials……….. 26

2.2 Performance of the frequency-dependent flow rate………. 28

2.2.1 A simple spring-mass-damper system under simple harmonic excitation ... ….30

2.2.2 Complex representations ... 34

2.3 Analysis of the simplified micropump for the flow rate……….. 35

2.4 The added mass and added damping……… 40

2.5 Analysis of the frequency shift with the interactions exerted on the micro-diaphragm………...43

2.6 The system dynamic for a peristaltic micropump……… 48 2.6.1 Electrical field ... 49 2.6.2 Mechanical field ... 53 2.6.3 Governing equations... 55 2.6.4 Flow field... 65 Chapter 3…...….68

Fabrication and Test...…68

3.1 Design and fabrication of a peristaltic micropump……….. 68

3.2 Experimental setup………... 72

3.3 Experiments and results……… 73

Chapter 4…...90

Results and Discussion ...90

4.1 Discussion and conclusion………... 90

4.2 Outlook………. 91

List of Figure

Fig. 1 A Typical Biochip Layout—Component ... 15

Fig. 2 Valveless rectification micropumps ... 19

Fig. 3 Tesla pump: (a) design example (b) rectification effect... 19

Fig. 4 Flow rate range of different pump principles. ... 20

Fig. 5 Actuation sequences of a peristaltic micropump based on piezoelectric actuation micropump... 24

Fig. 6 The flow chart of a six-phase driving peristaltic micropump... 25

Fig. 7 The deformation of a piezoelectric device when subject to an electrical voltage... 27

Fig. 8 A simplified diaphragm micropump viscously damped system with harmonic excitation. ... 31

Fig. 9 The schematic of a diaphragm micropump for analysis of flow rate 36 Fig. 10 Vibration of a beam of uniform cross section loaded at the middle with a block of weight W ... 41

Fig. 11 A simplified model for diaphragm micropump ... 45

Fig. 12 The illustration of frequency shift ... 48

Fig. 13 The general environment for the system ... 49

Fig. 14 Definition of the 3, 4 and6 phase durations... 50

Fig. 15 Actuation sequences of a 2-phase peristaltic micropump ... 51

Fig. 16 Definition of the 2-phase durations. ... 51

Fig. 17 A top view of the stresses obtained from FEM analysis in the 1st mode frequency... 54

Fig. 18 The stress distributed at different position in the 1st mode... 55 Fig. 19 a. Schematic of the cross section of a piezo-Pyrex-diaphragm

bi-layer b. Deflection of the pump diaphragm D1: diameter of PZT;

D2: diameter of Pyrex; S: distance between Pyrex and PZT edges;

diaphragm thickness t1 =191μm and t2 = 150 μm ... 58

Fig. 20 The solution algorithm for piezo-diaphragm-fluid coupled solver . 60 Fig. 21 The three layers model under 100Hz sinusoidal excitation (water, 100V). Grids at the center area of each layer are denser. ... 61

Fig. 22 Transient behavior of a point centered on diaphragm-fluid interface ... 62

Fig. 23 The photo of the finished diffuser/nozzle... 62

Fig. 24 The modal shape analysis for PZT and Si diaphragm bi-layer ... 65

Fig. 25 (a) Schematic of the peristaltic micropump (b) SEM photo of the step channel (c) Recirculation region ... 66

Fig. 26 Schematic representation of the rectangular channel ... 66

Fig. 27 Schematic of peristaltic micropumps... 69

Fig. 28 Glass etching process... 70

Fig. 29 Silicon etching process. ... 71

Fig. 30 The photomask of channel and chamber ... 71

Fig. 31 The complete peristaltic micropump ... 71

Fig. 32 The schematic of the experimental setup... 73

Fig. 33 Displacement of the middle moving diaphragm as a function of phase frequency at 100 Vpp(4P16)... 74

Fig. 34 Flow rate vs. phase frequency at 100 Vpp... 75

Fig. 35 Displacement of the middle moving diaphragm as a function of phase frequency at 100 Vpp (4P20)... 76

Fig. 36 Flow rate vs. phase frequency at 100 Vpp (4P20)... 77

Fig. 38 The displacement of diaphragm versus the driving frequency at 100

V(4P10)... 78

Fig. 39 The Flow rate of pumping versus the driving frequency at 100 V (4P10)... 79

Fig. 40 The displacement of diaphragm versus the driving frequency for different working fluids. ... 79

Fig. 41 The 3-phase sequence displacement frequency response versus the chamber height... 80

Fig. 42 The 4-phase sequence displacement frequency response versus the chamber height... 81

Fig. 43 The 6-phase sequence displacement frequency response versus the chamber height... 82

Fig. 44 4-phase three chambers driving ... 82

Fig. 45 4-phase the only middle chamber driving ... 83

Fig. 46 The charging and discharging loop of the differential amplifier... 84

Fig. 47 The displacement vs. frequency in 3-phase sequence by reducing the collector resistance from 50 K to 10 K Ohm... 85

Fig. 48 The flow rate vs. frequency in 3-phase sequence by reducing the collector resistance from 50 K to 10 K Ohm... 85

Fig. 49 The displacement vs. frequency in 4-phase sequence by reducing the collector resistance from 50 K to 10 K Ohm... 86

Fig. 50 The flow rate vs. frequency in 4-phase sequence by reducing the collector resistance from 50 K to 10 K Ohm... 86

Fig. 51 The displacement vs. frequency in 6-phase sequence by reducing the collector resistance from 50 K to 10 K Ohm... 87

Fig. 52 The flow rate vs. frequency in 6-phase sequence by reducing the collector resistance from 50 K to 10 K Ohm... 87

Fig. 53 The displacement vs. frequency operated in +80 V to -20 V, +50 V to -50 V and +20 V to -80 V deferential outputs ... 88 Fig. 54 The flow rate vs. frequency operated in +80 V to -20 V, +50 V to -50 V and +20 V to -80 V deferential outputs... 89

List of Table

Table 1 Mechanical pumping principles ... 17

Table 2 The comparison between cycle frequency and driving sequence... 51

Table 3 The cycle frequency factor... 53

Table 4 Properties of PZT-5H ... 56

Acknowledgement (in Chinese)

在交大兩年的碩士班生活中，首先要感謝陳俊勳老師，讓我順利的 完成研究所學業，在學術上提供一個跨領域的學習機會，讓我們可以 接觸不同領域的知識。在此亦感謝國立成功大學的張凌昇老師以及南 台科技大學的許藝菊老師在百忙之中給予本論文的批評與指正，使本 論文更趨完善，在此致上最高的敬意。 在研究的生活過程中，深受生命晶片實驗室成員的幫忙與指教，首 先感謝松儒以及垣杰在實驗及電路上提供不少建議與協助，以及在模擬 上相互成長的國華、耀文、偉雄與念暉。其次感謝宜良、明燦、瑭原、 智淵、鎵豪、南江、敏峰、浩凱、浩君與交大的同學嘉鴻、靜慈。還有 其它無法一一列入的研究夥伴，謝謝你們長期的陪伴與照顧，有你們的 陪伴，使得兩年的研究生活，增添許多樂趣與回憶。 更感謝的是我的父母，他們給我最大的支持，在我失落時給我精神 的鼓勵，讓我順利完成學業，無後顧之憂。僅以本研究獻給所有關愛我 的人。

CHAPTER 1

INFINITESIMAL MACHINERY: AN

INTRIGUED WORD

1.1 Introduction

Is there plenty of room at the bottom? A brilliant idea was presented by Richard P. Feynman on December 26, 1959, at the annual meeting of the America Physical Society at the California Institute of Technology [1]. The problem seems like a good idea in

theory, but in practice it has recently proved that there is a strong

trend for miniaturization using integrated circuit (IC) fabrication

process, such as etching, photolithography, deposition, bonding and so on. This trend results on one hand from the fact that small

components and system perform differently. Micro-electro

-mechanical system (MEMS) has been opened new thrusts into the world and make it possible to fabricate small size devices and systems with high functionality, precision, and performance. Based on these characteristics, MEMS devices and systems have found some applications, such as automobile, aerospace, communication, medical, etc.

Recently there has been enormous interest in the research and development of microfluidics because of increasing demands from biological applications such as genomics, proteomics, and drug

discovery. Therefore, an important direction in the development of analytical techniques is toward microanalyzer, Generic names for these new micrometer-featured devices include “micro- total analysis systems” (μ-TAS) (Manz et al., 1990) [2], lab-on-a-chip (Colyer et

al.,1997 ﹔ Moser et al.,1995), biochip, or, simply “chip.” A

promising analytical tool for analyzing proteins and protein complexes in the biology laboratory of the future is a microfluidic device. These "laboratories" (as the Fig.1) are fabricated using photolithographic processes developed in the microelectronics industry to create circuits of tiny chambers and channels in a quartz, silica, or glass chip. They direct the flow of liquid chemical reagents just as semiconductors direct the flow of electrons. These reagents can be diluted, mixed, reacted with other reagents, or separated — all on a single chip. In some cases devices have been named based on their particular application, for example, PCR chip [4], gene chip, while for others the device is named for characteristic structure feature, for example, microspot or microarray. The drug delivery microsystem consists of micropumps, biosensors, flow sensors, microvalves, microreservoirs, microneedles, and a feedback controller.

Fig. 1 A Typical Biochip Layout—Component [3]

Micropump is one of the MEMS devices, which can be used for drug delivery applications. It provides the driving force to mobilize fluids in the system, which then enables functions such as mixing, reaction, injection, and separation, in other words, this device as the main part of a drug delivery system transfers the fluid (drug) from the drug reservoir to the body (tissue or blood vessel) with high performance, accuracy, and reliability. Most drugs have a range of concentrations of greatest efficacy in the body, above which they are toxic and below which they have no therapeutic benefit [5].Conventional drug delivery routes such as oral tablets or injections are not easily able to control the rate of drug delivery or the target area of the drug. Consequently, initial concentration of the drug in the blood peaks above the level of toxicity and then gradually decreases over time to an ineffective level and the patients

have to take the drug frequently. In order to control drug release better, drug delivery systems (DDS) are necessary. Small size and high precision of micropumps have made them useful for chemotherapy, insulin delivery for diabetic patient, and drug dosing for cancer patient and so on [6].

1.2 Literature Review

Research on micropumps has been popular among MEMS scientists for more than 20 years [7–11]. In the first years of research this popularity may have been caused by the fascinating fact that these tiny devices produce fluid motion that is visible to the naked eye. Now more and more applications drive researchers to improve

their micropumps. Most micropumps found today can roughly be

divided into two groups [12]:

“Continuous flow micropumps” are based on a direct transformation of nonmechanical or mechanical energy into a

continuous fluid movement. While mechanical pumping was mostly

used in macroscale pumps and micropumps with a relatively large size and high flow rates, this second category discovers its advantages in the microscope. Since the viscous force in microchannels increases in the second order with miniaturization, the first pump category can not deliver enough power to overcome the high fluidic impedance in the microscale.

principles by which mechanical energy is applied to the fluid. Under this system, mechanical pumps are divided into two major categories: displacement pumps and dynamic pumps.

In displacement pumps, energy is periodically added by the application of force to one or more moveable boundaries of any desired number of enclosed, fluid-containing volumes, resulting in a direct increase in pressure up to the value required moving the fluid through check valves or ports into the discharge line. Check-valve pumps, peristaltic pumps, valve-less rectification pumps, and rotary pumps belong to the displacement category.

In dynamic pumps, mechanical energy is continuously added to increase the fluid velocities within the machine. The higher velocity at the pump outlet increases the pressure. Centrifugal pumps and ultrasonic pumps belong to the dynamic category. (Table 1)

Table 1 Mechanical pumping principles

The so-called “displacement pumps or reciprocating micropumps” use the oscillatory or rotational movement of mechanical parts to

displace fluid. Micropump development has started with “piston type” reciprocating micropumps like micro diaphragm pumps and peristaltic micropumps that do still form the main representatives of this class in the MEMS world. Mechanical pumps can handle a large variety of fluids, but often involve complicated structures and present integration challenges, as is evident in rotary pumps [13] (requiring bearings) and some diaphragm pumps requiring check valves [14](it function under conditions of a small compression ratio and of high pump pressure), diffuser/nozzle [15] (or valveless rectification micropump) pumps (as Fig.2) Besides, a Tesla [16] Type Valve (as Fig.3) is one of the no-moving parts (NMP) type valves used in micropumps for Micro Electro Mechanical System (MEMS) devices. This pump type can be realized easily in silicon with DRIE technology. Its principle of operation is based on the rectification of the fluid flow. For the same pressure drop, the flow in the forward direction through the valve is greater than the flow in the reverse direction, thus if an oscillating flow field can be set up, there should be net mass flow in the forward direction.

Fig. 2 valveless rectification micropumps

Forward

Reverse

Fig. 3 Tesla pump: (a) design example; and (b) rectification effect

In addition, different methods of actuation have been applied to micropumps such as electrostatic actuators, piezoelectric actuators,

magnetohydro- dynamic, shape memory alloy actuators, and thermal actuators. However, most of them have complex structures and are difficult to be miniaturized. Furthermore, their output power is limited and not sufficient for practical applications. Piezoelectric actuators are most promising, nevertheless, owing to their simple

structure, and great output power density. Van Lintel is the first

person who used a piezoelectric disc glued to a glass membrane [17]. Under a voltage difference the piezoelectric disc changes its lateral dimension which results in a bending moment in the dimorph. The original prototype was able to produce a maximum pressure of 100 cm water and a maximum yield of 10 μl/min at 1 Hz block wave

actuation. Figure 4 illustrates the typical flow rate range of

micropumps. For flow rates more than 10 ml/min, miniature pumps

or macroscale pumps are the most common solutions. The typical operation range of displacement micropumps lies between 10 μl/min and several milliliters per minute. For flow rates less than 10 μl/min, alternative dynamic pumps or nonmechanical pumps are needed for an accurate control of these small fluid amounts.

There are many groups now active in the design of micropump. A simple analytic model is presented to optimize the valveless diffuser pump by Olsson et al [18-19] and Ullmann [20] analyzed the performances of single and double chamber micropumps and discussed the dependence of the flux on pressure difference between the inlet and the outlet.

Pan et al [21] investigated the mechanical properties of fluid–membrane coupling for a valveless micropump, but the unsteadiness of the flow field within the micropump was not

accounted for in their investigation. Two years further on, the

inertial effects [22] was presented to shows that a phase shift lagging the excitation force exists in the vibration response. The dynamic response of a cantilever is investigated to find the connections between the nature frequency and structure by Naik [23]. Kan [24] expounds the concept of dynamic response into piezoelectric cantilever-valve micropumps. The analysis results suggest that check efficiency of cantilever valve depends on phase shift, which increases with the increasing of driving frequency. Several authors have considered low-order or “electrical-equivalent” models for micropumps.

Among all kinds of pumps, peristaltic pumps have many

advantages for biomedical applications. First of all, the peristaltic

micropump doesn’t have a moving part. The main advantage of the peristaltic pump is that no seals, valves or other internal parts ever touch the fluid. Due to the simplicity of their structures, peristaltic

pumps have found many applications in the pharmaceutical,

chemical, and food industries. Besides this, the action of a peristaltic

pump is very gentle, which is important if the fluid is easily damaged. Therefore, the particles and living cells would not be stuck or damaged. The risk of clogging in the channels is also reduced,

as a result. Secondly, the pump could be operated in the forward

and backward directions. Furthermore, it doesn’t have passive check valves as flow directing elements that open and close frequently; therefore, the valves will not easily be worn out and

cause mechanical fatigue. In other words, peristaltic pumps are

good for long-term treatments. Especially, it is suitable for reagents and cells pumping. Finally, the planar design is less complex with high pump performance, and has the ability to pump a wide variety of fluids.

B Husband [25] presents the peristaltic micropump with three PZT actuated glass membranes and silicon channels, is integrated

within the

μ

TAS device with microfluidic reaction chambers.

1.3 Working Principle and Performance for a

Peristaltic Micropump

In contrast to diffuser/nozzle pumps, peristaltic pumps synchronize several piezo discs in a wave-like motion. This peristaltic motion transports the fluids in one direction and requires no diffuser/nozzle. Classical peristaltic pumps generate the

wave-like motion by a wheel with roles along the circumference. The roles press on a flexible silicon rubber tube and cause the wave-like motion when the wheel rotates [26].

In our design, the basic elements of the peristaltic micropump consist of three chambers, and to operate in a peristaltic motion by the driving circuit. The driving circuit actuates the PZT to produce the pressure on the diaphragm with oscillating motion. The common driving scheme of a peristaltic micropump was divided into six phases (100, 110, 010, 011, 001, and 000) as depicted in Fig. 1. In phases 1 to 3 fluids was drawn into the pump chamber through the inlet valve, which is then displaced through outlet valve in phases 4 to 6. Thus one circle is complete. To repeat in circles, make the pump achieve self priming.

Fig. 5 Actuation sequences of a peristaltic micropump based on piezoelectric actuation micropump

Below is a series of six-phase diagrams illustrating

Fig. 6 The flow chart of a six-phase driving peristaltic micropump

The driving scheme of the four-phase mode (100, 110, 011, and 001) was the six-phase scheme that removes the process of phases 3 and 6. By the same way, the driving scheme of the 3-phase (100, 010, 001) was the six phase scheme that removes the process of phases 2, 4 and 6.

CHAPTER 2

THE DAMPING EFFECTS ON

THE MICRO-DIAPHRAGM

2.1 Piezoelectric Materials

The piezoelectric force has been widely used for micromechanical devices. The effect was discovered by Jacques and Pierre Curie in 1880. They discovered that if special crystals were subject to mechanical tension, they became electrically polarized and the polarization was proportional to the extension. They also discovered that the opposite was true; if an electrical field was applied across the material it deformed. This is known as the inverse piezoelectric effect. Piezoelectricity involves the interaction between the electrical and mechanical behavior of the medium. To the first order this is described as [27]

{

S

(

T

,

E

)

}

6×1

=

s

6E×6

T

6×1

+

d

6×3

E

3×1 (2-1)

where S is the strain, is the compliance tensor under conditions

of constant electric field, T is the stress, d is the piezoelectric charge constant tensor and E is the electric field. The deformation of a piezoelectric crystal is illustrated in Fig. 6 In the absence of mechanical loads Eq. (2-1) gives

E

(2-2)

(2-3) And

Fig. 7 The deformation of a piezoelectric device when subject to an electrical voltage

where is elongation along the poling axis, l is the device length

along the poling axis, U is the electrical voltage,

l

Δ

a

Δ

is elongation

perpendicular the poling axis and a is the device length perpendicular to the poling axis. Normallyd33>0 and d31< 0.

Examples of piezoelectric materials are quartz, LiTaO3, PZT and ZnO. Non-piezoelectric materials, e.g. silicon, can be exited by

depositing a thin film of a piezoelectric material, e.g. PZT or ZnO. Another solution is to mount a piezoelectric disk on the non-piezoelectric material. This eliminates the problem of making the film thick enough that high voltages can be applied without dielectric breakdown (sparks/short circuits across the film). The piezoelectric effect can be used to bend a diaphragm, e.g. in a pump. The principle is illustrated in Fig. 7 where a piezoelectric disk is glued to a diaphragm. When a voltage is applied across the piezoelectric disc it deforms and forces the diaphragm to bend.

2.2 Performance of the Frequency-dependent Flow Rate

Small vibrational structures, typically in the shape of diaphragm、beam and plate at the micron scale fabricated by silicon technology, have received ever increasing interest because they can be used as key components in developing sophisticated microelectromechanical systems (MEMS) including microsensors and actuators.

In this section we will examine the responses of single-degree of–freedom systems. This is one of the most important topics to master, since the more complicated cases can be treated as if they are simply collections of a spring-mass system. Now, I would like to focus attention on one of the chambers in the peristaltic micropump for the sake of a simplification. With a micropump, the release rate of drug delivery is able to be controlled easily to maintain the

therapeutic efficacy. Hence, with taking into account the influence of liquid added mass and added damping on the dynamic characteristics of the micro diaphragm and actuator in the natural frequency, a performance of piezoelectric micropump was investigated for this purpose.

If a statically or periodically loaded elastic system, such as a spring、beam or membrane, is disturbed in some manner from its position of equilibrium, the internal forces and moments in the deformed configuration will no longer be in balance with the

external loads and vibrations will occur. In the piezoelectric

micropump, the fluid flow is driven by the vibrating actuator. At the same time, the fluid plays a key role in resistance to the actuator vibration. The actuator vibration, the micro-diaphragm movement and the fluid flow are thus coupled. The fluid reaction force is represented as an added mass and added damping contribution to the dynamic response of the structure without affecting their stiffness. The added mass and added damping depends on both the fluid density and viscosity, as well as on the gap height in the chamber and a solid surface. Therefore with fluid (e.g. air, water or blood) as the pump medium, the dynamical behaviors of the actuator and the micro-diaphragm are different from those in vacuum.

2.2.1 A Simple Spring-mass-damper System under Simple

Harmonic Excitation

The basic element in the micro-diaphragm pump is shown in Fig. 9. Therefore, very simplified, this can be seen as a Helmholtz resonator or as a mass-spring system. We define the bi-layer “piezoelectric disk and Pyrex glass diaphragm” as the micro-diaphragm. In this case, the spring is the micro-diaphragm. Such a system is illustrated in Fig. 8 where also viscous damping is

included. fe is the periodic actuating force and P is the dynamic

pressure exerted on the diaphragm by the liquid. Clearly, undamped

systems are not highly accurate models of the real world (especially for high viscosity fluid) since we know that the unforced oscillations of real systems always decay away eventually. Like all systems possessing mass and elasticity it is capable of free vibration, i.e. it has natural frequencies. When such a system is subject to harmonic excitation it is forced to vibrate at the same frequency as the excitation frequency. When a system is excited by a transient force, the resulting excitation take place at the natural frequencies of the system with the amplitude varying depending on the excitation.

Fig. 8 A simplified diaphragm micropump viscously damped system with harmonic excitation.

In consideration of the liquid added mass and added damping, the dynamic equation of a damping driven harmonic oscillation of the micro-diaphragm can be expressed as

)

(t

f

kx

x

c

x

m

_&&

+ &

+

=

(2.4) Dividing by m, as we did for solving ordinary differential equation

x

&&

+

_m

c

x

&

+

_m

k

x

=

f

(t

)

(2.5)

The parameters m, c, k and F stand for the total effective mass of

the micro-diaphragm in liquid environment (M=Mv +Mad ,Mv is the

mass of the micro-diaphragm and Mad is the adding mass of liquid),

the added damping constant, the spring constant of the micro-diaphragm and the driving force provided by the piezoelectric actuator, respectively. We next assume that our forcing is sinusoidal

and introduce the notations ﹕ 2 ,

m k w_n = mk c 2 = ζ

we obtain 2 2 sin(wt) m f x w x w x&&+

ζ

_n&+ _n = (2.6) which is the differential equation of motion for forced vibrations with viscous damping. A particular solution of Eq. (2.3) can be taken in the form (2.4) because we realize that sines and cosines are linearly independent functions﹔there is no way to get one from the other. wt N wt M x = cos + sin (2.7)

where M and are constants. To determine these constants, we

use operator to solve it

N ) ( ) ( 1 _{1 0} 1D a D a x r t a D a n n n n + + + + = − − L , namely L(D)x=r(t) ) sin( ) 2 ( 2 2 _wt m f x w D w D +

ζ

_n + _n = (2.8) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + = iwt n n m e m f w D w D I t x _{2 2} 2 1 ) (

ζ

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + = ⇒ m f w w w i iw e I t x n n iwt m _{( )}2 ₂ 2 1 ) ( ζ

⎥

⎦

⎤

⎢

⎣

⎡

+

−

+

=

m

f

w

w

i

w

w

wt

i

wt

I

t

x

n n m

₍

₎

₂

_ζ

sin

cos

)

(

_{2 2}

Upon multiplying both numerator and denominator by the term
(w_n2 ₋w2)₋i2ζw_nw

[

]

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ + − − − + = _{2 2 2} 2 2 ₂ ) 2 ( ) ( 2 ) ( ) sin (cos ) ( w w w w w w i w w wt i wt I t x n n n n m _ζ ζ

wt w w w w m f w w wt w w w w m f w w t x n n n n n n sin ) 2 ( ) ( ) ( cos ) 2 ( ) ( ) 2 ( ) ( _{2 2 2 2} 2 2 2 2 2 2 _{ζ ζ} ζ + − − + + − − = (2.9)

Thus, M and are ﹕ N

2 2 2 2 _{) (2 )} ( ) 2 ( w w w w m f w w M n n n ζ ζ + − − = and _{2 2 2 2} 2 2 ) 2 ( ) ( ) ( w w w w m f w w N n n n ζ + − − = (2.10)

Only can ever equal zero and this occurs only if the forcing

frequency is equal to the system’s natural frequency. At all frequencies

N

M is nonzero, and hence we always will have some

finite response. The total solution of Eq. (2.6) is obtained by adding the general solution to the particular solution (2.9)

[

w t

]

c

[

w t

]

c t x( ) exp ( 1) _n exp ( 2 1) _n 2 2 1 − + − + − − − = ζ ζ ζ ζ wt w w w w m f w w wt w w w w m f w w n n n n n n sin ) 2 ( ) ( ) ( cos ) 2 ( ) ( ) 2 ( 2 2 2 2 2 2 2 2 2 2 _{ζ ζ} ζ + − − + + − − + (2.11)

The first two terms in Eq. (2.11) represent damped free vibrations,

that due to the factor the free vibrations gradually subside,

leaving only the steady forced vibrations represented by the last two terms. These forced vibrations are maintained indefinitely by the action of the disturbing force from the excitation of PZT actuators and, therefore, are of great practical importance. We shall now see how they affected by damping.

nt e−

We’ll use the trigonometric formula﹕

)

sin(

)

cos(

)

cos(

)

sin(

)

sin(

a

−

b

=

a

b

−

a

b

[

sin( )cos

φ

cos( )sin

φ

]

sin

coswt N wt A wt wt

2 2 _N M A_d = + and tan 1( ) b a − − =

φ

The steady-state response Eq. (2.9) may be written in the equivalent phase angle form as

)

sin(

)

(

t

=

A

wt

−

φ

x

_{d (2.12)}

from which, and substituting the actual values of M and N

)

2

(

tan

1 _{2 2}

w

w

ww

n n

−

=

−

ζ

φ

_(2.13) and 2 2 2 2 _{) (2 )} ( 1 w w w w m f A n n d ζ + − = _(2.14)

In this case

φ

has a physical meaning, that is, the amount that the

output lags the input.

Introducing the nondimensional frequency﹕

n w w = Ω

tan

(

₁

2

2

)

1

Ω

−

Ω

=

−

ζ

φ

_(2.15) and _{2 2 2} ) 2 ( ) 1 ( 1 Ω + Ω − = ζ m f A_{d (2.16)}

2.2.2 Complex Representations

Although representing a system’s response as Eq. (2.7) is physically attractive, it is a bit awkward mathematically. Our

solution to this problem will be to represent as complex notation, namely ) (t x n n

_m

e

iwt

f

x

w

x

w

x

+

₂

+

2

=

&

&&

ζ

(2.17)

The steady-state solution is given by

iwt

e

x

x

=

, and n n

w

i

ww

w

m

f

x

ζ

2

)

(

1

2 2

₋

₊

=

in terms of massm, springk constant and damping constantc,

respectively

w

k

c

i

w

k

m

k

f

x

+

−

=

2 2 2

1

1

(2.18)

2.3 Analysis of the Simplified Micropump for the

Flow Rate

The diaphragm structure acts as a “piston” which transfers energy to the fluid in the pump chamber, when driving voltage is sine wave, in one period, the fluid flows into the pump chamber in [0, T/4] and [3T/4, T]. The fluid flows out the pump chamber in [T/4, 3T/4]. So it is just necessary to calculate the flow rate in [T/4, 3T/4]. The Mach number is the ratio between the flow velocity u and the speed of sound a and is given by:

a

V

M

_a

=

_(2.19)

The Mach number is a measure of the compressibility of a gas and can be thought of as the ratio of inertial forces to elastic forces. For

the Mach-number in the pump chamber is approximately 5.25e-5

(less than 0.3). Under the conditions the density of a gas will not change significantly while it is flowing through a system. The flow is then considered to be incompressible even though fluid, a gas, is still considered compressible. The compression could be neglected when calculating the pump’s flow rate. The analysis of an incompressible gas flow is greatly simplified as it can be treated with the same versions of the governing equations that will be derived for liquid flows.

Fig. 9 The schematic of a diaphragm micropump for analysis of flow rate

Assume hv is the distance between diaphragm and the bottom of

pump chamber as the figure 9. It could be expressed as[28]﹕

wt

h

h

_v

=

′

+

δ

_dia

sin

2

π

(2.20)

where δdia is the centerline displacement of the interior surface of the

membrane, i.e. the vibration amplitude ﹔ h’ is the diaphragm’s original distance between diaphragm and pump chamber’s bottom surface﹔w is vibration frequency

Introducing a parameter to correct Eq. (2.20), The membrane shape

factor

γ

﹕ it was also determined based on the static membrane

displacement ﹕ dia c A

δ

ν

γ

= _(2.21)

where

ν

is the volume swept out by the membrane, is the area

of the chamber

c

A

Not considering the compression and leakage, the instantaneous volume (V) of the pump chamber in [T/4，3T/4] is:

BL

wt

h

V

=

(

′

+

γδ

_dia

sin

2

π

)

(2.22) where L and B is the length and width of the pump chamber. The change ratio of the instantaneous volume is equal to the

instantaneous flow Q, we have:

wt

BL

w

V

Q

=

′

=

2

π

γ

δ

_dia

cos

2

π

(2.23) As we known, the pump’s instantaneous flow also could be

expressed as:

A

=

Bh

=

B

(

h

′

+

γδ

dia

sin

2

π

wt

)

(2.24)

where A and v are the instantaneous micro-diaphragm and chamber cross-section area and the flow rate in pump chamber length

direction (x direction), respectively; From the eq. (2.23), and eq. (2.24), we have:

wt

h

wt

w

L

A

Q

v

dia dia

π

γδ

π

π

γ

δ

2

sin

2

cos

2

′

+

⋅

=

=

_(2.25)

Thus, the mean flow rate v is﹕

∫

∫

+ ′ ⋅ = = 3 4 4 1 4 3 4 1 2 sin 2 cos T T dia dia T T dt wt h wt L T w T dt v v π γδ π γ δ π (2.26) The eq. (2.26) can be simplified to:

dia dia

h

h

T

L

v

γδ

γδ

+

′

−

′

=

ln

2

(2.27)

Assume

η

is leakage factor, the mean flow rate considered the

leakage is as follow: dia dia h h T L v γδ γδ η + ′ − ′ = ln 2 (2.28)

The fundamental relation for the piezo effect will be described by the following equation:

t U d sT l l 31 + = Δ (2.29)

Here Δl_{l is the relative expansion of the piezoelectric material}

achieved by applying a voltage U between the diaphragm and the upper-electrode (t: thickness of the piezo ceramic). The

electromechanical ransformation depends on the piezoelectrical

coefficient of the ceramic material, described by d31, on the stiffness

s and on the mechanical load T

This is especially noteworthy in the case of a combination of a rectangular piezo disk and rectangular Pyrex glass diaphragm, the load T cannot be deduced by analytic equations. For a approach, the calculation of the pump diaphragm displacement is possible

assuming circular plates with a radius Rp instead of square plates for

the diaphragm and piezo disk. As an essential result of this model the deflection w(r) of the pump diaphragm can be calculated by

_⎟⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − = 2 2₂ 2 31 ₁ 3 ) ( p p R r R t U d r w b (2.30)

For r=0 the maximal deflection w(r) can be calculated to

2 2 31 max 3 ) 0 ( _p b R t U d w r w = = =− _(2.31)

From the Eq. (2.16), (2.29) and (2.31), the diaphragm’s amplitude could be expressed as 2 0 2 2 0 2 31 2

2

1

3

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

=

w

w

w

w

t

U

d

l

b dia

ζ

δ

(2.32)

U d l w w w w t h U d l w w w w t h Lw v b b 31 2 2 0 2 2 0 2 2 31 2 2 0 2 2 0 2 2 3 2 1 3 2 1 ln 2

γ

ζ

γ

ζ

η

+ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − ′ − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − ′ = _(2.33)

This is the expression of the diaphragm pump’s mean flow rate which relates with the amplitude and frequency of the voltage, pump’s chamber length, bimorph’s parameters, leakage factor, the membrane shape factor etc..

2.4 The Added Mass and Added Damping

A primary concern in the design of the peristaltic micropump is damping effects on the frequency response. In certain simplification cases, the vibration problems can be reduced to the case of a system with one degree of freedom. In addition, the mass of the spring, beam or membrane can be neglected in comparison with the mass of

the load weight . In our case, therefore, we can regard the load

weight as excitation force from the piezoceramic, PZT. Although

these simplifications are accurate enough in many practical cases, there are technical problems in which a detailed consideration of the accuracy of such approximation becomes necessary. With the miniaturization of the size (range of

W W

m

μ to ) in the

microelectromechanical systems (MEMS) technology, the analysis of vibration and damping play important role in mechanical

engineering design. When a micro-diaphragm vibrates in a viscous fluid, the fluid offers resistance to the motion of the diaphragm. The fluid loading can be interpreted as the sum of two forces: an inertial force that is proportional to the acceleration of the diaphragm (the proportionality constant is called added mass), and a viscous or dissipative force that is proportional to the velocity of the diaphragm (the proportionality constant is called added damping or viscous damping coefficient).

In order to determine the effect of such simplification on the frequency of vibration, an approximate method developed by Lord Rayleigh will now be discussed. In applying this method some assumption regarding the configuration of the system during vibration must be made. The frequency of vibration will then be from a consideration of the conservation of energy in the system.

Fig. 10 vibration of a beam of uniform cross section loaded at the middle with a block of weight W

wl

of the beam is small in comparison with the load W，it can be assumed with sufficient accuracy that the deflection curve of the beam during vibration has the same shape as the static deflection

curve for a concentrated load at the middle. is the displacement

at the center of the beam (see Fig.10) and we express the displacement of any element located at s distance x from the support as : m y

)

4

3

(

₃ 3 2

l

x

xl

y

y

=

_m

−

The maximum kinetic energy of the beam itself will be

g

y

wl

dx

l

x

xl

y

g

w

_m m l

2

35

17

)

4

3

(

2

2

2 2 3 3 2 2 0

&

&

−

=

∫

This kinetic energy of the vibrating beam must be added to the

energy Wym _g

2

2

& _{of the loaded concentrated at the middle in order to} estimate the effect of the weight of the beam on the period of

vibration. In this case the period of vibration will be the same as for a massless beam loaded at the middle at the middle by the weight W W wl 35 17 + = ′ _(2.34)

Considering a homogeneous beam, which is vibrating in liquid, has

thickness h, length L, width B, mass density ρ_dand the mass per unit

length m_d =hbρ_d. In such a simple model, the added mass and added

3 0 3 0 3 , 10 h B c h B

m_ad = ρ = μ with

ρ

and

μ

standing for the liquid density

and the liquid dynamic viscosity, respectively.

Lc

C

m

m

L

M

(

_{d ad}

)

,

12

π

35

17

₊

₌

=

2.5 Analysis of the Frequency Shift with the interactions

exerted on the micro-diaphragm

With a micropump, such diaphragm structure acts as a “piston” to provide power for the handling of microliter-scaled fluid volumes desired in many lab-on-a-chip chemical and biomedical applications. In the design of the mechanical efficiency, the pump performance hangs on using resonance to generate sufficient motion of the diaphragm. We are concerned with the interactions between a system and its environment (such as the implicit pressure and shear stress at the solid-fluid interface and the force exerted by the PZT actuator) for the influence on output resonance frequency.

The micro-diaphragm (the Piezoelectric disk and Pyrex diaphragm bi-layer) is integrated with surrounding walls; therefore, it can be considered as a flexible double-clamped beam spring-mass-damper system (as Fig.8). However, the various working fluids play different roles in resistance to the diaphragm vibration. Many inputs to physical systems are periodic in nature. For example, the forces exerted on marine structures by ocean

waves, the acoustic and electric waveforms of music and speech and mechanical vibrations exerted on structures due to unbalanced elements are all inherently cyclic or periodic in nature. Now consider a simple spring-mass-damper system under harmonic excitation in stead of the real input signals (block wave actuation) that will be taken up in the next section (as the Fig.14 on page 50). Therefore, a second-order system with an input-output differential equation can obtained

M

y

&&

+

ky

=

F

sin(

wt

)

−

F

chamber (2.34)

where M, k are mass, spring constant, respectively. The spring is the micro-diaphragm.

The actual input driving signals can be closely approximated by sinusoidal waveforms. Any physical periodic phenomena may be represented by an infinite sum of harmonically related sinusoids, and therefore knowledge of the system frequency response to a sinusoidal input provides a basis for determining the response to a broad class of periodic inputs. Now, the core of the problems is how to define the flow resistance exerted on the micro-diaphragm. We assume viscous liquid at volume flow rate, Q, is pumped through the central diaphragm and the narrow gap between the parallel disks (as the Fig.11) [29]. The flow rate is low, so the flow is laminar and the pressure gradient due to convective acceleration in the gap is negligible compared to the gradient due to viscous forces. We assume that the velocity profile at any cross section in the gap is the same as for fully developed flow between stationary parallel

plates. Here the flow is axisymmetric and therefore it is most convenient to take the control volume as annular ring. It is of length, , and has circumference,dr 2πr.

Fig. 11 A simplified model for diaphragm micropump

Therefore, the pressure gradient,dp_dr, as a function of radius is

3 0 12 1 2 r h p r Q ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ − = π h r Q dr dp 3 0 6 π μ − = (2.35)

And the flow rate Q can be approximate estimated

Q =

π

(a2 −r2)&y (2.36)

From (2.11), we know the solution is the sum of a

homogeneous component and a particular solution ) (t y ) (t y_p

∑

(2.37) =

+

=

n i p t i

e

y

t

C

t

y

i 1

)

(

)

(

λ

Where n is the system order (for our case n =2),

λ

iare the roots

constants to be determined from the initial conditions. The solution to the homogeneous part decays exponentially with time and is only initially significant. The particular solution is a steady state oscillation of the same frequency as the excitation and it can be assumed to be of the form, w the angular frequency (rad/sec)

y

=

A

d

sin(

wt

−

φ

)

,

y wA

&

=

d

cos( )

wt

(2.38)

Now substitute the result (2.36) into Eq. (2.35) and integrate to find the pressure distribution

⎟

⎠

⎞

⎜

⎝

⎛

₋

₊

+

=

2 2 2 3 0 0

2

1

2

1

ln

6

a

r

a

r

a

h

y

P

P

μ

&

_,

₍

_{r <r <a}

₎

0 (2.39)

The force acted on the micro-diaphragm is given by

0

( )2 a ch

F =

∫

p r πrdr

Thus the flow resistance can be evaluated

⎭

⎬

⎫

⎩

⎨

⎧

−

−

⎥⎦

⎤

⎢⎣

⎡

₋

⎥⎦

⎤

⎢⎣

⎡ +

+

=

2 2 0 2 0 2 0 4 3 0 0 2

₍

₎

8

1

2

1

)

ln(

)

(

2

1

4

1

12

r

a

a

r

a

r

a

h

y

p

a

F

_ch

π

πμ

&

(2.40) The viscous damping constant is proportion to the micro-diaphragm

velocity,wA_d cos( )wt ; therefore we can find the damping constant.

(2.41)

Thus, the main four parameters： the fluid viscosityμ, the chamber

height , the micro-diaphragm length a, the geometric constants h₀

) ln( 0

a r

model for diaphragm micropump on the damping effects. Substitute the Eq. (2.41) into (2.34) can obtain

0 2

)

sin(

wt

a

p

F

ky

y

c

y

M

_&&

+ &

+

=

−

π

(2.42) Introduce the notations：

M k w_n2 = _， mk c 2 = ς

We know that the magnitude of the transfer function for a forced-excited system is given by

2 2 2 2 0 2

)

2

(

)

(

1

)

(

n n d

ww

w

w

m

p

a

F

w

g

A

ς

π

+

−

−

=

=

_(2.43) To determine the frequency at which our amplitude-response curve is

a maximum(which we will call the damped oscillation

frequency, )，we need only differentiate with respect to and set

the result equal to zero . Thus, we obtain [30]

d

w

w

Mk

c

w

w

w

_{d n n} 2 2

₁

2

1

−

=

−

=

ς

_(2.44) where n k w M

= _{is the natural frequency of the micro-diaphragm in}

vacuum and 2

C Mk

ζ = _{is the damping factor which is drawn on to}

modify the natural frequency of the actuating diaphragm in vacuum. Hence, a damped frequency of the micro-diaphragm can be obtained

with varying damping factor for light damping )

2 1

(ς ≤ . From Eq.

undamped natural frequency , with the shift increasing toward zero as the damping is increased. We define the difference between the damped and nature frequency as the frequency shift. In the following chapter “fabrication and test “I shall be examining the phenomena of frequency shift through experiments.

n

w

Fig. 12 The illustration of frequency shift

2.6 The System Dynamic for a Peristaltic Micropump

The coupled fluid–structure–electric interaction problem may be considered as a three field problem, i.e. fluid flow, structural deformation and the electric field (see Fig. 14). The effectiveness of a system is intrinsically related to its dynamic behavior. The governing equations are: Piezoelectric constitutive equation, spring-mass-damper systems and Navier–Stokes equation.

Fig. 13 The general environment for the system

2.6.1 Electrical Field

We introduce the phase durations for j = 1. . . 6, as defined in

figure 13. The actuation sequences of a 3, 4, and 6 driving phase

can refer to figure 5. The cycle duration T is the sum of all . We

also introduce the relative duration of each phase

j T j j T t = _T . The

driving scheme is therefore defined by all and the total duration T.

The driving frequency is defined as

j t

1

Fig. 14 Definition of the 3, 4 and6 phase durations.

By comparison with the driving signals of the valveless rectification micropump (diffuser/nozzle), we design the so called “2-phase sequence” driving signals. The three chambers have no signals phase lag. We will discuss the resonance frequency peak shift due to different phase sequence later.

Fig. 15 Actuation sequences of a 2-phase peristaltic micropump

Fig. 16 Definition of the 2-phase durations.

Table 2 The comparison between cycle frequency and driving sequence

To investigate the pump performance of micropump, the pump was actuated with the operation frequency, i.e. the phase frequency. Furthermore, the cycle frequency means the oscillation times of the micro-diaphragm through a period. Table 2 illustrates the value of

cycle frequency at different driving sequence. As a example of the

operation (phase) frequency 120Hz, the cycle frequency of 2, 3, 4, and 6 phase sequence are 60 , 40, 30 and 20 times respectively. It reveals the oscillation frequency of 3-phase sequence is higher than those by the 4-phase and 6-phase sequences during the same time interval. Since sinusoidal waveforms are used as the basis for representing other periodic and transient waveforms through the process of Fourier synthesis, it seems reasonable to draw an analogy between 2-phase sequence and sinusoidal waveforms for explaining the movement of micro-diaphragm. Consequently, the cycle

frequency factor

α

presented in Table 3 is drawn on to modify 3, 4,

Table 3 The cycle frequency factor

Therefore, Using Eq. (2.38), (2.41), (2.44) and Table.3, we should

consider the cycle frequency factor

α

to correct Eq. (2.41) into

(2.45) 4 2 2 0 0 0 3 4 0

12

1 1

1

1

( )

ln( )

(

)

4 2

2

8

r

r

a

c

a

h

a

a

a

α

⋅

πμ

⎧

⎡

⎤ ⎡

⎤

_r

2 2

⎫

=

_⎨

_⎢

+

_{⎥ ⎢}

−

_⎥

−

−

_⎬

⎣

⎦ ⎣

⎦

⎩

⎭

(2.45)

Where

α

3−phase

=

0.444

,

α

4−phase =0.25 and

α

6−phase

=

0.167

2.6.2 Mechanical Field

In the section, the geometrical dimensions of the actuator and the diaphragm and the chamber height are design which can produce more displacement without complicated fabrication, leading to a higher expansion/compression ratio.

Alternating voltage causes the PZT component to expand and contract along the horizontal direction. This induces a bending stress on the diaphragm, which in turn pumps the fluid through the

chamber. From the simulations below, we turn up the interesting facts that the maximum stresses occur near the edge of the

diaphragm where the largest bending moments

exist (Fig. 19 and 20).

) 012 . 0 01 . 0 ( <r< m

To assure pump reliability for high cycle fatigue, it is, therefore, necessary to design this pump so that the maximum stress level is kept lower than the stress endurance limit of the diaphragm material. This requirement is vital for many types of micro devices considering the role micro pumps play in sustaining the reliability of MEMS for biomedical applications, such as lab-on-a-chip devices. Consequently, we can take the concept of reliability analysis (lifetime, operating time and electric load of a piezoelectric actuator micropump) into consideration further in the future.

Fig. 17 A top view of the stresses obtained from FEM analysis in

Fig. 18 The stress distributed at different position in the 1st mode

2.6.3 Governing Equations

A piezoelectric patch is utilized as the actuator. PZT-5H, a special type of piezoelectric, is used in the simulation. Properties of PZT-5H are given in table1. The coupled electro-mechanical

constitutive equation for the actuator is

(2.46)

where

ε

_kl is the mechanical strain tensor,

σ

_ij is the mechanical

stress tensor, Ek is the electric field vector, eijk is the piezoelectric

constant tensor; is the elastic stiffness constant tensor at

constant electric field and it is a 6 × 6 symmetric tensor. E

ijkl

C

The Pyrex diaphragm in the peristaltic micropump is integrated with surrounding walls; therefore, it can be considered as a clamped plate as the Figure 19. The governing equation of forced vibration of a thin clamped plate is

P f t W h W D _Pyrex = _e − ∂ ∂ + ∇4

_ρ

2 ₂ (2.47) where (2.48)

and fe is the periodic actuating force, which can be solved from

equation (2.46), E is the elastic modulus of the Pyrex diaphragm,

λ

is Poisson’s ratio of the Pyrex diaphragm, ρPyrex is the density of

the Pyrex diaphragm, h is the thickness of the Pyrex diaphragm, and

P is the dynamic pressure exerted on the diaphragm by the liquid. To

solve equation (2.47), P should be solved from the Nervier–Stokes

equations at every time step. Since we assume the diaphragm to be a

clamped plate, displacements, curvatures and velocities of the clamped plate at edges should be zero. Therefore, the boundary

conditions for a clamped plate can be written mathematically as (2.49)

(2.50)

Fig. 19 a. Schematic of the cross section of a piezo-Pyrex-diaphragm bi-layer b. Deflection of the pump

diaphragm D1: diameter of PZT; D2: diameter of Pyrex; S:

distance between Pyrex and PZT edges; diaphragm

thickness t1 =191μm and t2 = 150 μm

Properties of the fluid used in this simulation are listed in table 5. Since the characteristic length of the micropump is of the order of

10−6 _{and the Reynolds number is very low, the flow can be}

considered as an incompressible laminar flow, which can be described using the Navier–Stokes equations (2.52) and the mass continuity equation (2.53)

(2.53)

Table 5 Fluid properties

In this study, the commercial software CFD-ACE+ is used to solve the piezo-diaphragm-fluid coupling for multidisciplinary analysis. The solution algorithm for full coupling is shown in Fig.21.The broad approach is conventional in that the fluid and the structure are solved sequentially. The model considered is shown in Fig. 22. with three layers: a square PZT stack which is easier to manufacture than a corresponding circular multi-layer stack, silicon diaphragm and chamber with two outlets. In the design of the model, the silicon diaphragm is fully covered by a PZT layer and the chamber is fully covered by the PZT–silicon diaphragm bi-layer. Fig. 23.(a)(b) indicate the dynamic analysis result of the vertical displacement and pressure of a point centered on diaphragm-fluid interface respectively. The deflections at different actuating frequencies are presented in Table 6.